What is s(t) if s'(t)=36t^5+4t^3?
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Given the derivative of the function s(t) , we have to find s(t). For this we integrate the derivative of the function.
s(t) = Int [ s'(t)]
=> s(t) = Int [ 36*t^5 + 4*t^3]
=> s(t) = Int [ 36*t^5] + Int [ 4*t^3]
=> s(t) = 36* t^6 / 6 + 4*t^4 / 4 + C
=> s(t) = 6*t^6 + t^4 + C
The function s(t) = 6*t^6 + t^4 + C
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What is s(t) if s'(t)=36t^5+4t^3.
If s'(t) = 36t^5+4t^3, then s(t) = Int s'(t) dt.
=> s(t) = Int (36t^5+4t^3 )dt.
=> s(t) = Int 36t^5 dt + Int 4t^3 dt.
=>s(t) = 36 (1/6) t^6 +4(1/4)t^4+ C.
s(t) = 6t^6 +t^4 +C .
Therefore if s'(t) = 36t^5+4t^3, then s(t) 6t^6+4^4 +C, where C is a constant of integration.
According to the rule, s(t) could be determined evaluating the indefinite integral of s'(t)
Int (36t^5+4t^3)dt
We'll apply the additive property of integrals:
Int (36t^5+4t^3)dt = Int (36 t^5)dt + Int (4 t^3)dt
We'll re-write the sum of integrals, taking out the constants:
Int (36t^5+4t^3)dt = 36 Int t^5 dt + 4Int t^3 dt
Int (36t^5+4t^3)dt = 36*x^6/6 + 4*x^4/4
We'll simplify and we'll get:
Int (36t^5+4t^3)dt = 6x^6 + x^4 + C
The function s(t) is: s(t) = 6x^6 + x^4 + C
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